Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 391786, 13 pages

http://dx.doi.org/10.1155/2015/391786

## Effect of Variable Fluid Properties on Natural Convection of Nanofluids in a Cavity with Linearly Varying Wall Temperature

^{1}Department of Mechanical Engineering, Faculty of Engineering, University of Malaya, 50601 Kuala Lumpur, Malaysia^{2}Institute of Mathematical Sciences, Faculty of Science, University of Malaya, 50601 Kuala Lumpur, Malaysia^{3}UTM Centre for Industrial and Applied Mathematics, Department of Mathematical Sciences, Universiti Teknologi Malaysia, 81310 Johor Bahru, Johor, Malaysia

Received 16 October 2014; Revised 24 January 2015; Accepted 25 January 2015

Academic Editor: Quanxin Zhu

Copyright © 2015 M. Bhuvaneswari et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The present study analyzed convective heat transfer and fluid flow characteristics of nanofluid in a two-dimensional square cavity under different combinations of thermophysical models of nanofluids. The right vertical wall temperature is varying linearly with height and the left wall is maintained at low temperature whereas the horizontal walls are adiabatic. Finite volume method is used to solve the governing equations. Two models are considered to calculate the effective thermal conductivity of the nanofluid and four models are considered to calculate the effective viscosity of the nanofluid. Numerical solutions are carried out for different combinations of effective viscosity and effective thermal conductivity models with different volume fractions of nanoparticles and Rayleigh numbers. It is found that the heat transfer rate increases for Models M1 and M3 on increasing the volume fraction of the nanofluid, whereas heat transfer rate decreases for Model M4 on increasing the volume fraction of the nanoparticle. The difference among the effective dynamic viscosity models of nanofluid plays an important role here such that the average Nusselt number demonstrates an increasing or decreasing trend with the concentration of nanoparticle.

#### 1. Introduction

Convective heat transfer cooling enhancement technique has been a major problem in engineering and technological applications. In the past, several researchers have conducted research on convective heat transfer technique in various aspects to get better heat transfer medium. Recently, fluids with nanometer sized particles suspended into them called nanofluids are considered in research. Nanofluids are dilute colloidal suspensions containing nanoparticles. The base fluids may be ethylene glycol, refrigerant, water, oil, or mixtures of one or more liquids. The nanoparticles used nanofluids made of metals, metal oxides, carbides, nitride, and carbon. Nanofluids are mainly used as coolant in heat transfer equipment such as heat exchangers, electronic cooling system, and radiators. The novel properties of nanofluids make it potentially useful in many applications, including microelectronics, fuel cells, pharmaceutical processes, hybrid-powered engines, engine cooling, vehicle thermal management, domestic refrigerator, chiller, grinding, machining, and boiler flue gas temperature reduction.

Since detailed investigations in all respects have been required before using the nanofluids in practical applications, we studied theoretically to find the heat transfer and fluid flow characteristics of nanofluid inside a differentially heated cavity under different combinations of thermophysical models of nanofluids in the present study. There are several models used for calculation of physical properties of nanofluid by several researchers [1–3]. Considerable amount of experimental and numerical studies has been conducted for the effect of improvement of the thermal conductivity of nanofluids because most of the researchers are thinking that thermal conductivity plays a dominant role in convective transport [4–6]. However, less number of studies is devoted to the effect of viscosity of the nanofluids. Since the viscosity of the fluid is increased when nanoparticles are suspended into them, we must consider the effect of viscosity on convective flow and heat transfer.

Khanafer et al. [7] investigated buoyancy driven convection heat transfer in a two-dimensional enclosure using nanofluids numerically. They found that heat transfer rate increases with an increase in the nanoparticle volume fraction. Hwang et al. [8] theoretically studied the thermal characteristics of natural convection of water based Al_{2}O_{3} nanofluids in a rectangular cavity. They showed that the ratio of heat transfer coefficient of nanofluid to that of base fluid is decreased as the size of nanoparticle increases. Ho et al. [9] numerically studied the effects of uncertainties in the effective viscosity and thermal conductivity of nanofluids on convection in enclosure. They concluded that the effective dynamic viscosity should be taken into account when studying the heat transfer efficacy for natural convection in enclosures. Jou and Tzeng [10] performed a numerical study for the effect of small aspect ratio of enclosures on natural convection of nanofluids in a rectangular enclosure. They found that the average Nusselt number at the hot wall is increased as the aspect ratio decreases. Santra et al. [11] studied numerically natural convection of nanofluids in a differentially heated square cavity. They treated the nanofluid as non-Newtonian. Sivasankaran et al. [12] numerically studied the free convection of nanofluids with different nanoparticles in a square cavity with linearly varying wall temperature. They found that the increment in average Nusselt number strongly depends on the nanoparticle chosen. Rashmi et al. [13] numerically studied natural convection of nanofluids using computational fluid dynamics approach. They found that heat transfer decreases on increasing the particle volume fraction. Sivasankaran and Pan [14] numerically investigated natural convection of nanofluids in a cavity with sinusoidal temperature distributions on vertical sidewalls. It is observed that the heat transfer rate is enhanced by nonuniform temperature distributions of both walls as compared to the case of uniform heating on one wall. Cianfrini et al. [15] numerically studied the free convection of nanofluids in a partially heated cavity. They found that the heat transfer decreases as the nanoparticle size, the width of the cavity, and the length of the heater are increased. Bouhalleb and Abbassi [16] numerically examined the natural convection of nanofluid in an inclined rectangular cavity heated from one side and cooled from the ceiling. They observed that heat transfer increases first and then decreases on increasing the inclination of the enclosure for all aspect ratios.

In addition, most of the previous studies investigate the convection heat transfer with either uniform wall temperature or uniform heat flux thermal boundary conditions. However, these thermal boundary conditions are not suitable in many practical applications such as heat exchangers, injection moldings, and solidification processes. The effect of temperature dependent properties of water near its density maximum on convection is studied numerically by Sivasankaran and Ho [17]. They found that the temperature difference parameter did not create any significant effect on the heat transfer. Sivasankaran et al. [18] analyzed the magnetoconvection of cold water in an open cavity with temperature dependent properties. They found that the Nusselt number behaves nonlinearly with Marangoni number.

In most of the numerical or experimental studies on convection heat transfer of nanofluids, researchers have considered the effect of physical properties; particularly, the thermal conductivity of nanofluids on natural convection in enclosures and very few investigations for viscosity are found in the literature. There are many models for thermophysical properties of nanofluids based on experimental data or theoretical results available in the literature. In most of the studies on convection heat transfer in cavities, researchers take any one of the models and investigate the heat transfer and fluid flow characteristics inside the cavity. Very few studies only concentrate on comparing the models of thermophysical properties of nanofluid.

Also, we need the knowledge of the effect of variable wall temperature on some applications. Therefore, the present study aims to investigate the effect of different models of effective viscosity and thermal conductivity of Al_{2}O_{3} nanofluid on natural convection in a cavity with linearly varying wall temperature.

#### 2. Mathematical Analysis

Consider a two-dimensional square cavity of length filled with nanofluids as shown in Figure 1. The cavity is heated differentially between the two vertical sidewalls at different temperature. The left wall temperature is varying linearly with height. The horizontal walls are assumed to be adiabatic, nonconducting, and impermeable to mass transfer. The nanofluid in the enclosure is a solid-liquid mixture with uniform volume fraction , shape, and size of nanoparticles (Al_{2}O_{3}) dispersed within base fluid water and it is also Newtonian. The flow is assumed to be incompressible and laminar. It is assumed that both nanoparticles and base fluid are in thermal equilibrium. The Boussinesq approximation is valid in the buoyancy term and all other thermophysical properties are assumed to be constant. In addition, the viscous dissipation is assumed to be negligible.